Optimal solution of a class of non-autonomous initial-value problems with unknown singularities

In this paper we are interested in a rigorous analysis of the solution of a class of initial-value problems with singularities. We consider scalar non-autonomous IVPs with separated variables and one unknown singularity in each variable (which leads to four unknown ‘events’ in the two-dimensional space). Many algorithms proposed in the literature for singular IVPs are practically oriented. They do not avoid heuristic arguments, and are often checked for efficiency by numerical experiments. We design an adaptive algorithm with no heuristic steps for solving the considered problems, and provide rigorous bounds on the error. We show that in spite of the presence of singularities the algorithm preserves the (optimal) error known for the regular case. Lower bounds on the error of any algorithm in the case of a larger number of singularities are also discussed.